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\begin{equation*}p_k = \sum_{a=0}^k C(2k+1, 2a+1) \frac{\sqrt{2}}{\pi} \left(a - \begin{matrix} \frac{1}{2} \end{matrix} \right)! 
{\left(a + g + \begin{matrix} \frac{1}{2} \end{matrix} \right)}^{- \left( a + \frac{1}{2} \right) } e^{a + g + \frac{1}{2} }\end{equation*}
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